MODELLING OF ROAD VEHICLE-BRIDGE INTERACTION USING A COMPOUND MULTIBODY/FINITE ELEMENT AP- PROACH

Transcription

1 23 rd International Congress on Sound & Vibration Athens, Greece July 2016 ICSV23 MODELLING OF ROAD VEHICLE-BRIDGE INTERACTION USING A COMPOUND MULTIBODY/FINITE ELEMENT AP- PROACH Ľuboš Daniel, Ján Kortiš University of Žilina, Faculty of Civil Engineering, Department of Structural Mechanics, Žilina, Slovakia lubos.daniel@fstav.uniza.sk Bryan Olivier, Georges Kouroussis University of Mons UMONS, Faculty of Engineering, 7000 Mons, Belgium When a vehicle runs on a bridge, dynamic effects cannot be longer neglected compared to the static deflection either on highway bridges or railroad bridges. Although dynamic interaction between a moving vehicle and the sustaining bridge can be studied in the frequency domain using simple and complementary techniques (e.g. modal superposition), a time domain approach remains interesting because it offers more possibilities in the dynamic analysis: nonlinear dynamic studies, stability criteria, analysis of local defects and/or distributed pavement roughness, The paper presents a method developing a coupled multibody/finite element approach with application to truck problems. To take into account the bounce and the pitch motion of the vehicle, a multibody model of the truck has been created using the in-house framework EasyDyn. Presented as a C++ library, this framework provides to the user the numerical construction and integration of the equations of motion, from the kinematics, expressed according to the minimal coordinates approach, and the forces exerted on the mechanical system. The bridge is built according to a finite element approach and integrated to the vehicle model using Fromm's model in order to take into account the mechanics of pneumatic tyres and the tyre/bridge interaction. The numerical results are compared to existing analytical solutions and to experimental data. It turns out that the proposed approach is efficient with the advantage of versatility. 1. Introduction The growing request to increase the transport capacities that ensure a safe and an effective transportation of people and goods leads to the extension of the road and rail network. With the creation of this extra capacity, the field of structural vibration is now an important engineering challenge due to higher speeds and heavier loads, especially around specific buildings and sensitive structures [1]. It also leads to increase the demands for the maintenance of these structures. These issues are also related to the building of new bridge structures. The dynamic behaviour of bridges including vehicle-structure interaction is an important issue. Two kinds of dynamic analysis are usually performed. In one hand, the free vibration of a beam provides interesting information about the natural frequencies of a bridge, offering a way to validate the bridge model in the frequency domain [2]. The other analysis is the dynamic response of bridge traversed by vehicles, to represent the transient bridge response. Various possibilities exist to model the vehicle: a simple moving load (or a sequence of moving loads) travelling the bridge at constant 1

2 speed [3], a single degree of freedom system [4] or a complete multibody model representing the different moving parts of the vehicle, connected by the suspensions [5,6]. The choice of any of these vehicle model results from a compromise between the mathematical modelling difficulties, the expected results accuracy, and the computational burden [7]. Because of the complicated mathematical considerations in analytical formulation, numerical simulations are preferred. Moreover, the availability of an analytical solution requires some simplified hypotheses. The finite element method (FEM) is the most widely used numerical method and it is usually used with commercial software packages. However, FEM is not an appropriate method to define the vehicle behaviour. An alternative is the use of multibody simulation for the vehicle, coupled to the bridge finite element model through co-simulation techniques [8]. This induces however an important complexity directly visible on the modelling approach and on the integration scheme, but such kind of models could be studied also in the framework of nonlinear dynamics including dynamic dampers [9,10]. In consideration of the aforementioned aspects, an alternative is proposed in this paper, based on the approach described in [11]. The application case is based on road vehicle moving on a bridge structure. A multibody model has been created using the in-house framework EasyDyn [12,13], providing to the user the numerical building and integration of the equations of motion, from the kinematics, expressed according to the minimal coordinates approach, and the forces exerted on the studied mechanical system. A 2-D finite element of the bridge is then added to the model, using an analytical force relationship for the tire-structure contact. Analytical solution and field tests are then used to validate the proposed approach. 2. Position of the problem The selected case of this study is based on a real bridge structure and vehicle. Figure 1 shows the studied bridge which has a length of 87 m and has three spans. Eight bearing girders are used, composed of prefabricated prestressed concrete structures. The bearing girders in each span have the length of the span. The girders from the neighbour spans are not connected together and act independently as a simple supported beam. The middle span with a length of 29 m is chosen to be inspected. Experimental tests have been performed in June 2015, considering the response of the bridge traversed by heavy lorry. The results of the test are presented as measured deflections in the middle of the span. Regarding the problem, it can be formulated and solved in a single plane. A recent analysis showed that, in the present case, a solution obtained with a simple 2D model is relatively close to solutions of more complex 3D models [14]. The only restriction lies in the fact that the vehicle has to follow the path of the middle of the road, which implies that the torsional effects cannot be covered. Figure 1: (left) View of the bridge structure located between Varín and Mojš (Slovakia) and (right) the position of the car on the bridge during the experimental tests. 2 ICSV23, Athens (Greece), July 2016

3 3. Using EasyDyn framework EasyDyn has been developed by the Department de Theoretical Mechanics, Dynamics and Vibrations of the University of Mons, initially devoted to education but successfully used afterwards for research purposes [12]. The library is written in C++ and provides 4 modules: the vec module, introducing classes related to vector calculus: vectors, rotation tensors, inertia tensors, and homogeneous transformation matrices; the sim module to integrate second-order differential equations; the mbs module, a frontend to sim which automatically builds the differential equations of motion of a multibody system from the kinematics and the applied forces; the visu module, allowing to build scenes composed of moving objects and directly usable with the associated animation tool EasyAnim. Its purpose is to help users to set up and to integrate the equations of motion of a multibody system with a minimal effort. These several modules allow (Fig. 2): the generation of bodies kinematics from the position information using a symbolic tool called CaGeM, the numerical building of equations of motion, from these kinematics and the forces applied on the mechanical system, the integration of these equations written in residual form, the animation of the scenes, for a better visualization of the motion. A detailed description of the framework can be found in [12,13]. with CAGeM (symbolic) or manually kinematics of each body (T 0,i,v i, a i, ω i, ω i ) kinematics (v i, a i, ω i, ω i ) ComputeMotion() vec Forces (R i and M Gi ) AddAppliedEfforts() mbs Differential equations f(q, q, q,t)=0 + any other systems definition ComputeResidual() sim sim visu with any graphing utility EasyDyn library with EasyAnim Figure 2: Multibody simulation dataflow [12]. 3.1 Vehicle multibody model The approach with minimal coordinates [15] is used to build a system of ordinary differential equations for the vehicle. The configuration parameters are chosen, in a number corresponding to the number of degrees of freedom (denoted by ). This approach has the advantage to eliminate automatically all the joint forces and allows an efficient integration with the existing numerical ICSV23, Athens (Greece), July

4 techniques. If the system comprises bodies, only equations can be built around the configuration parameters. The application of the d Alembert s principle leads to the following equations of motion,, Φ Φ 0 1,, (1) with and Φ the mass and the central inertia tensor of body i, and the resultant force and moment, at the centre of gravity Gi, of all applied forces exerted on body i, the acceleration of the centre of gravity of body i,, the partial contributions of in the velocity of the centre of gravity of body i and, the partial contributions of in the angular velocity of body i. The 2D model of the vehicle moving over the bridge is based on the lorry T815 (Fig. 3). The geometrical and inertial data are provided in [16]. The position and orientation of each body (car body, bogie, wheelset, or other inertial component, with a role in the vehicle dynamics) are gathered in a homogeneous transformation matrix, which gives the situation of the associated local frame with respect to another frame as a function of the configuration parameters. For any complex mechanical system, the motion of each body can be decomposed into a succession of elementary motions (rotation about one axis, pure translational displacement) defined as successive multiplications of simple matrices. To obtain the kinematics and the partial contributions of Eq. (1), only differentiation operations on terms of homogeneous transformation matrices are necessary. This is performed using the symbolic tool CaGeM. Suspensions, as well as wheel/bridge contacts, are defined as nonlinear force elements applied to specific locations of bodies, using dedicated routines. Figure 3: Model of the lorry Tatra T Bridge finite element model The shape of girders as well as their position are showed in Figure 4, which displays also the layers of the road and the cornices with handrails. Figure 4: Cross-section of the studied bridge. 4 ICSV23, Athens (Greece), July 2016

5 The bridge consists of a two-dimensional finite element model. A simply supported Euler- Bernoulli beam elements is employed to define the model of the bridge structure. It is assumed that the bridge properties are constant and the load is symmetric along the bridge width. The material parameters are constantly defined along the bridge by the Young s modulus ( =28 GPa) and the shape of the bridge cross-section is characterized by the geometrical moment of inertia in vertical plane ( =2.7 m 4 ) and area of the cross-section ( = 9 m 2 ). The unit mass of the bridge is assumed to be = kg.m -1. The number of elements depends on the grade of discretization. In the present case, one hundred elements of a length of 0.29 m are used. The interpolation formula is based on the Hermitian cubic shape functions that meet the continuity requirement. They are conveniently expressed in terms of the dimensionless coordinate of a beam element (2) 2 (3) 3 2 (4). (5) The nodal forces and torques applied on the beam element i are given by /, (6) /, (7) /, (8) /, (9) where /, represents the wheel/bridge contact force acting on the selected beam. The employment of the previous assumption leads to assemble the mass and stiffness matrices. The damping characteristic is considered as Rayleigh damping defined by two parameters α = s -1 and β = s. The values of the damping characteristics were obtained from a previous experimental analysis of the bridge [14]. The associated damping matrix is therefore given by. (10) The complete FEM of the bridge is implemented on EasyDyn through the corresponding mass, damping and stiffness matrices built from its mechanical and geometrical properties. 3.3 Tyre/bridge contact: Fromm's model In order to understand how the vehicle interacts with the road, the tire model was implemented in EasyDyn. The tire element is considered as a force relationship based on the Fromm's model [17]. The distribution of vertical pressure in the contact zone is parabolic along the circumference so that a tyre could be compared to an assembly of springs in the longitudinal, radial and lateral directions as shown in Figure 5. Using Eqs. (2) to (9), the vertical position of the beam at the contact point is calculated from the four degrees of freedom of the beam element where the vehicle contact is located. This way allows avoiding the creation of moving nodes for the coupling between the vehicle and the bridge. Defects on the road surface, corresponding to a local unevenness and/or an overall road irregularity, can be defined by any kind of deterministic function or measured profile. This information is included directly in the tyre force definition, allowing a full coupling between the multibody subsystem (vehicle) and the finite element subsystem (bridge), defining the coupled motion of the vehicle/bridge system from point to point according to the vehicle speed. ICSV23, Athens (Greece), July

6 Figure 5: Tire equivalent spring modelling. 4. Validation Two validation cases are presented in this section: a comparison of bridge deflection with an analytical formula in order to validate the bridge finite element model, the steady-state response and the contact law, and a comparison with experimental data for validating the whole system, including the dynamic effect of the vehicle. 4.1 Analytical solution The numerical approaches presented in the paper are compared with typical bridge analytical solutions. The used analytical solution describes the moving load as a force moving over a simply supported beam of length. The beam response for any distance and time with the assumption of only light damping is given by the approximated formula [18], sin sin (11) where = 0.04 describes the effect of the speed, = 0.1 rad.s -1 is the circular frequency of damping of the beam, is the deflection of the beam at mid-span, defined by and the natural circular frequency at the n-th mode of vibration of the simply supported beam (12). (13) The correspondence between the circular angular frequency and the loading speed is given by. (14) The results of Figure 6 show that both analytical solution and numerical approaches give very close results. The curve of the analytical solution is smooth in contrast with the curves provided by the numerical simulation. It is the result of the application of constant moving forces only, in contrast to the model of moving vehicle. The dynamic characteristics of the vehicle are neglected in Eq. (11). It can be also concluded that the bridge deflection can be well reproduced by analytical solution if the road surface is without significant imperfection. 4.2 Experimental data One of the most important steps in the process of evaluation of the numerical solution is to verify its results with the response of the real structure exposed to the similar loading. In order to quantify the dynamic effect of a moving truck on a bump, and to validate the proposed numerical model, an obstacle similar to a speed bump is considered. The vehicle speed is assumed to be constant at 8.5 km/h at when it was riding across the middle span. The response of the bridge was measured 6 ICSV23, Athens (Greece), July 2016

7 with the relative deformation sensor BOSCH. The measured arms of sensors were connected with the bottom surface of the bridge through wires. The measurement was performed in two specific places in the middle of the span. The sensor BOSCH1 measured the deflection in the middle of the cross-section and the sensor BOSCH2 measured the deflection at the edge of the cross-section. The output of the sensor BOSCH1 is compared with the results of the corresponding numerical model simulated under EasyDyn. Figure 7 compares both results and shows a good agreement between the two curves. The choice of a low vehicle speed offers the way to visualize the impact of each tyre on the local defect Displacement [m] Analytical solution Using EasyDyn Time [s] Figure 6: Bridge dynamic defection due to the passing of a lorry T815: comparison between the analytical solution and the results from EasyDyn simulation ( = 10 m/s) Measurement Using EasyDyn Displacement [m] Figure 7: Bridge dynamic defection during the passing of the lorry T815 over a bump at speed of 8.5 km/h: comparison between the results from experimental test and from EasyDyn simulation. 5. Conclusion Time [s] The analysis of the bridge vibration subjected to the passing of a truck can be performed by using different approaches. Since this kind of vehicle generates high dynamic loads, a compound numerical model was proposed in this paper, combining a multibody model for the vehicle and a FEM approach for the bridge. A particular attention was paid on the tyre/road contact by implementing Fromm s model and including any kind of road irregularity. The whole model was built using the in-house framework EasyDyn, which offers some advantages on the modelling development (including the possibility to implement specific routines) and on the versatility of the simulation code. The two proposed validation cases show that the model well represents the physical phenomena and therefore it is able to further research in the field. ICSV23, Athens (Greece), July

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